We consider the $k$-min-sum-radii ($k$-MSR) clustering problem with fairness constraints. The $k$-min-sum-radii problem is a mixture of the classical $k$-center and $k$-median problems. We are given a set of points $P$ in a metric space and a number $k$ and aim to partition the points into $k$ clusters, each of the clusters having one designated center. The objective to minimize is the sum of the radii of the $k$ clusters (where in $k$-center we would only consider the maximum radius and in $k$-median we would consider the sum of the individual points' costs). Various notions of fair clustering have been introduced lately, and we follow the definitions due to Chierichetti, Kumar, Lattanzi and Vassilvitskii [NeurIPS 2017] which demand that cluster compositions shall follow the proportions of the input point set with respect to some given sensitive attribute. For the easier case where the sensitive attribute only has two possible values and each is equally frequent in the input, the aim is to compute a clustering where all clusters have a 1:1 ratio with respect to this attribute. We call this the 1:1 case. There has been a surge of FPT-approximation algorithms for the $k$-MSR problem lately, solving the problem both in the unconstrained case and in several constrained problem variants. We add to this research area by designing an FPT $(6+\epsilon)$-approximation that works for $k$-MSR under the mentioned general fairness notion. For the special 1:1 case, we improve our algorithm to achieve a $(3+\epsilon)$-approximation.

翻译：本文研究带公平约束的$k$-最小和半径（$k$-MSR）聚类问题。$k$-最小和半径问题是经典$k$-中心与$k$-中位数问题的混合形式。给定度量空间中的点集$P$及数值$k$，目标是将点划分为$k$个聚类，每个聚类设有一个指定中心。需要最小化的目标函数是$k$个聚类半径之和（在$k$-中心问题中仅考虑最大半径，在$k$-中位数问题中则考虑各点成本之和）。近年来研究者提出了多种公平聚类概念，本文采用Chierichetti、Kumar、Lattanzi与Vassilvitskii[NeurIPS 2017]的定义，要求聚类构成需遵循输入点集在给定敏感属性上的比例分布。针对敏感属性仅有两个可能取值且在输入中等频出现的简化情形（称为1:1情形），目标是计算所有聚类在该属性上均保持1:1比例的聚类方案。近期$k$-MSR问题的FPT近似算法研究涌现，已解决无约束情形及多种约束变体问题。我们通过设计适用于前述通用公平概念的FPT $(6+\epsilon)$近似算法推进该领域研究。针对特殊1:1情形，改进算法可获得$(3+\epsilon)$近似比。